The regularized problem (REF )-(REF ) with \(C\in {\mathcal {M}}^{n \times n}\) 
has a
unique solution, say \(P_{\lambda }\)  [1]}. Indeed, the existence and uniqueness
follows from the boundedness of \(U ({\bf f}, {\bf g})\)  and the strict convexity of
the negative entropy.
On the one hand, as \(\lambda \)  increases, the unique
solution \(P_{\lambda }\)  converges to the solution with maximum entropy within the set
of all optimal solutions of the original Kantorovich's problem; see Proposition 4.1 in [2]},
\(P_{\lambda } \stackrel{\lambda \rightarrow \infty }{\longrightarrow } \underset{P \in \lbrace  P \in U({\bf f}, {\bf g}), \,\langle P, C \rangle = T_{C}({\bf f}, {\bf g})\rbrace }{\mathrm {argmin}} -H(P).\) 
